Recall that by the end of the last post, we had given a complete strategy for the (slightly degenerate) stochastic control problem that modelled how to select a photo from a photobooth. The strategy was determined by an infinite sequence, defined inductively by:
At the very end of the post, I gave a selection of highly non-rigorous justifications for why
for large .
After trying a bit more carefully, I think I can show the following:
More precisely, eventually
where , so in particular .
To set this up, just to avoid having ‘2’s everywhere, we rescale the s by 1/2, so that the recursive definition is now:
I will show that for suitably large , (though in practice this won’t be very large at all)
It will then suffice to show that the left hand side of this deduction holds for some .
To show that
it suffices to verify that:
When , we can bound using the difference of two squares:
A similar calculation, which is relatively straightforward to perform and a bit of a hassle to write up into WordPress, gives the corresponding upper bound. It remains to check that
and a direct computation (I used MATLAB) confirms this.